buttord

Syntax

Description

buttord calculates the minimum order of a digital or analog Butterworth
filter required to meet a set of filter design specifications.

Digital Domain

[n,Wn] = buttord(Wp,Ws,Rp,Rs) returns the lowest order, n, of the
digital Butterworth filter with no more than Rp dB of passband ripple and at least Rs dB of attenuation in the stopband. The scalar (or vector)
of corresponding cutoff frequencies, Wn, is also
returned. Use the output arguments n and Wn in butter.

Choose the input arguments to specify the stopband and passband
according to the following table.

Description of Stopband and Passband Filter
Parameters

Parameter

Description

Wp

Passband corner frequency Wp, the
cutoff frequency, is a scalar or a two-element vector with values between 0 and 1, with 1 corresponding to the
normalized Nyquist frequency, π radians per
sample.

Ws

Stopband corner frequency Ws, is a
scalar or a two-element vector with values between 0 and 1, with 1
corresponding to the normalized Nyquist frequency.

Rp

Passband ripple in decibels.

Rs

Stopband attenuation in decibels. This value is the number
of decibels the stopband is down from the passband.

If your filter specifications call for a bandpass or bandstop
filter with unequal ripple in each of the passbands or stopbands,
design separate lowpass and highpass filters according to the specifications
in this table, and cascade the two filters together.

Analog Domain

[n,Wn] = buttord(Wp,Ws,Rp,Rs,'s') finds
the minimum order n and cutoff frequencies Wn for
an analog Butterworth filter.
You specify the frequencies Wp and Ws similar
those described in the Description of Stopband and Passband Filter
Parameters table
above, only in this case you specify the frequency in radians per
second, and the passband or the stopband can be infinite.

Examples

Lowpass Butterworth Filter

For data sampled at 1000 Hz, design a lowpass filter with no more than 3 dB of ripple in a passband from 0 to 40 Hz, and at least 60 dB of attenuation in the stopband. Find the filter order and cutoff frequency.

Wp = 40/500;
Ws = 150/500;
[n,Wn] = buttord(Wp,Ws,3,60)

n =
5
Wn =
0.0810

Specify the filter in terms of second-order sections and plot the frequency response.

Bandpass Butterworth Filter

Design a bandpass filter with a passband from 60 to 200 Hz with at most 3 dB of passband ripple and at least 40 dB attenuation in the stopbands. Specify a sampling rate of 1 kHz. Have the stopbands be 50 Hz wide on both sides of the passband. Find the filter order and cutoff frequencies.

More About

Algorithms

buttord's order prediction formula
is described in [1]. It operates
in the analog domain for both analog and digital cases. For the digital
case, it converts the frequency parameters to the s-domain
before estimating the order and natural frequency, and then converts
back to the z-domain.

buttord initially develops a lowpass filter
prototype by transforming the passband frequencies of the desired
filter to 1 rad/s (for lowpass and highpass filters)
and to –1 and 1 rad/s (for bandpass and bandstop
filters). It then computes the minimum order required for a lowpass
filter to meet the stopband specification.